ATHENS—Greece moved a step closer to the end of its bailout regime on Monday after eurozone finance ministers approved the completion of Greece’s bailout review and agreed to disburse the next tranche of bailout funds in the coming weeks.

The Eurogroup meeting in Brussels endorsed Greece’s progress in reforms, as the country’s parliament voted on dozens of fiscal, labor and energy reforms last week, but said they the next slice of bailout cash worth €6.7 billion will be disbursed early February, after Athens settles some final...